Question

An elastic string of unstretched length ${}^{\prime }{L}^{\prime }$ and force constant ${}^{\prime }{k}^{\prime }$ is stretched by a small length ${}^{\prime }{x}^{\prime }$. It is further stretched by small length ${}^{\prime }{y}^{\prime }$. The work done in the second stretching is

• A. $\frac{1}{2}K{y}^2$
• B. $\frac{1}{2}K(x^2+y^2)$
• C. $\frac{1}{2}Ky(2x+y)$
• D. $\frac{1}{2}k(x+y{)}^2$

Answer 1

The correct option is B $\frac{1}{2}K(x^2+y^2)$

In the string elastic force is conservative in nature.
$\therefore W=-△U$
Work done by elastic force of string.
$W=-(U_F-U_i)=U_i-U_F$
$W=\frac{1}{2}k{x}^2-\frac{k}{2}(x+y{)}^2$
$=\frac{1}{2}k{x}^2-\frac{1}{2}k(x^2+y^2+2xy)$
$=\frac{1}{2}k{x}^2-\frac{1}{2}k{y}^2-\frac{1}{2}k{x}^2-\frac{1}{2}k\left(2xy\right)$
$=-kxy-\frac{1}{2}k{y}^2$
Therefore, the work done against elastic force
$W_{external}=-W=\frac{ky}{2}(2x+y)$.